1. **State the problem:** Simplify the expression $\left(3-\ln e\right)^3$. 2. **Recall the properties of logarithms:** The natural logarithm of $e$ is 1, i.e., $\ln e = 1$. 3. **Substitute the value:** Replace $\ln e$ with 1 in the expression: $$\left(3-\ln e\right)^3 = \left(3-1\right)^3$$ 4. **Simplify inside the parentheses:** $$\left(3-1\right)^3 = 2^3$$ 5. **Calculate the cube:** $$2^3 = 8$$ **Final answer:** $8$